Chapter 03.doc

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Chapter 3 The Time Value of Money 67

Chapter 3 The Time Value of Money

Solutions to End-of-Chapter Problems

Future Value of a Lump Sum

P3-1. You have $1,500 to invest today at 7 percent interest compounded annually.a. How much will you have accumulated in the account at the end of the following

numer of years!1. "hree years#. i% years&. 'ine years

. (se your findings in part )a* to calculate the amount of interest earned in1. the first three years )years 1 to &*#. the second three years )years + to 6*

&. the third three years )years 7 to *c. -ompare and contrast your findings in part )*. %plain why the amount of interestearned increases in each succeeding &/year period.

A3-1. uture alue2 n 3 4 )1 r*nor n3 4 )r,n*

a. 1. & 3 4 )1.07*& . 1. nterest earned 3 &8 4

& 3 $1,500 )1.##50+* nterest earned 3 $1,9&7.57

& 3 $1,9&7.57 /1,500.00$ &&7.57

#. 6 3 4 )1.07*6 #. nterest earned 3 68 &

6 3 $1,500 )1.5007&* nterest earned 3 $#,#51.10

6 3 $#,#51.10 /1,9&7.57$ +1&.5&

&. 3 4 )1.07* &. nterest earned 3 8 6

3 $1,500 )1.9&9+6* nterest earned 3 $#,757.6

3 $#,757.6 /#,#51.10$ 506.5

c. "he fact that the longer the investment period the larger the total amount of interestcollected is not une%pected and is due to the greater length of time that the principal

sum of $1,500 is invested. "he most significant point is that the incremental interestearned per & year period increases with each suse:uent &/year period. "he totalinterest for the first & years is $&&7.57, however, for the second & years )from year & to6* the additional interest earned is $+1&.&. or the third &/year period the incrementalinterest is $506.1. "his increasing change in interest earned is due to compounding,the earning of interest on pervious interest earned. "he greater the previous interestearned the greater the impact of compounding.

"his edition is intended for use outside of the (.. only, with content that may e different from the (.. dition. "his may not eresold, copied, or distriuted without the prior consent of the pulisher.


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69 Instructors Manual

Present Value of a Lump Sum

P3-2. You ;ust won a lottery that promises to pay you $1 million e%actly 10 years from today. investments during the 10/year period!1. 6 percent#. percent&. 1# percent

. ?ewor> part )a* under the assumption that the $1 million payment will e received in15 rather than 10 years.

c.


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Future Value of Cash Flow Streams

P3-4. Bim dwards and -hris 4hillips are oth newly minted &0/year old C* eginning ne%t month, while -hris intends to

invest $#,000 per month, ut he does not plan to egin investing until 10 years after Bimegins investing. * plan averages a1# percent annual return compounded monthly. =ho will have more +01)>* money atretirement!

A3-4. BimDs future retirement account at age 67 )r3 .1#E1# 3 .01F n =&7yrsG1#mosEyr3 +++ mos*2

&7 3 $1,000 )1.01*+++8 1I 3 $9,1#,596

.01

-hrisD future retirement account at age 67 )r 3 .1#E1# 3 .01F n3 #7yrsG1#mosEyr 3 +mos*2

&7 3 $#,000 G )1.01* +8 1 I 3 $+,9#5,##0

.01-learly, Bim will have far more money at retirement )$9,1#,596* than will -hris )$+,9#5,##0*.

P3-5. ?oert =illiams is considering an offer to sell his medical practice, allowing him to retirefive years early. He has een offered $500,000 for his practice and can invest this amountin an account earning 10 percent per year, compounded annually. f the practice is e%pectedto generate the following cash flows, should ?oert accept this offer and retire now!

End of Year Cash Flow

1 $150,000

2 150,000

3 125,0004 125,000

5 100,000

A3-5. on original retirement date if early retirement is chosen2

$500,000 )1.10*5 3 $905,#55

on retirement date if early retirement is not chosen2

$150,000 G)1.10*+ 3 $#1,615

$150,000 G)1.10*& 3 1,650$1#5,000 G)1.10*# 3 151,#50

$1#5,000 G)1.10*1 3 1&7,500

$100,000 G)1.10*0 3 100,000

$909,015

?oert =illiams should not retire early ecause the future value of his cash flows at theend of five years would e aout $&,000 less than if he continued wor>ing.



4/32


P3-6. or the following :uestions, assume an annual annuity of $1,000 and a re:uired return of1#percent.

a. =hat is the future value of an ordinary annuity for 10 years!. f you earned an additional yearDs worth of interest on this annuity, what would e the

future value!c. =hat is the future value of a 10/year annuity due!d. =hat is the relationship etween your answers in parts )* and )c*! %plain.

A3-6. a. A10 3 $1,000 G )1.1#*10 8 1 I 3 $17,5+

0.1#

. $17,5+ G)1.1#* 3 $1,655c. A10)annuity due* 3 $1,000 G )1.1#*10 8 1 I G)1.1#* 3 $1,655

0.1#

d. "he answers to parts and c are identical, implying that the future value of annuity due is simply the future value of an ordinary annuity plus an additional interest payment.

P3-7. ?oert ers a two/month paid saatical every sevenyears. ?oert, who ;ust started wor>ing for the firm, plans to spend his saatical touringurope at an estimated cost of $#5,000. "o finance his trip, ?oert plans to ma>e si%annual end/of/year deposits of $#,500 each, starting this year, into an investment accountearning 9 interest.

a. =ill ?oertDs account alance at the end of seven years e enough to pay for his trip!

. uppose ?oert increases his annual contriution to $&,150. How large will his

account alance e at the end of seven years!

A3-7. a. 3 #,500 G A )7, 9* 3 $#,500 G 9.##9 3 $##,&07. "herefore, ?oertDs alancewill not e enough for him to cover the trip.

. 3 &,150 G 9.##9 3 $#9,107. n this case the account alance will e enough for?oert to ma>e the trip.

P3-. Jina -oulson has ;ust contracted to sell a small parcel of land that she inherited a fewyears ago. "he uyer is willing to pay $#+,000 at closing of the transaction or will pay theamounts shown in the following tale at the beginningof each of the ne%t five years.


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3 6,000

4 8,000

5 10,000

a. =hat is the future value of the lump sum at the end of year 5!. =hat is the future value of the mi%ed stream at the end of year 5!c. e!d. f Jina could earn 10 percent rather than 7 percent on the funds, would your

recommendation in part )c* change! %plain.

A3-. a. 5 3 4 G)1.07*5

53 $#+,000 G)1.+0&*53 $&&,661



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7# Instructors Manual

. %e&"nn"n& of 'um(er of

)ear )ears *t+ FV , CFt*1 ./7+t Future Value

1 5 $ #,000 G1.+0& 3 $ #,905.10# + $ +,000 G1.&11 3 5,#+&.19& & $ 6,000 G1.##5 3 7,&50.#6+ # $ 9,000 G1.1++ 3 ,15.#05 1 $10,000 G1.070 3 10,700.00

"otal 3 $&5,#57.7+

c. Jina should select the stream of payments rather than the upfront $#+,000.

d. Lump sum53 4x )1.10*

5

53 $#+,000x )1.611*53 $&9,65#.#+

Mixed stream

%e&"nn"n& of 'um(er of

)ear )ears *t+ FV , CFtG*1 .1/+t Future Value

1 5 $ #,000 G1.611 3 $ &,##1.0## + $ +,000 G1.+6+ 3 5,956.+0& & $ 6,000 G1.&&1 3 7,96.00+ # $ 9,000 G1.#10 3 ,690.005 1 $10,000 G1.100 3 11,000.00

"otal 3 $&7,7+&.+#

'ote that, although the future sums of each alternative are larger at 10 than at 7, the10 upfront payments result in greater future value at the end of year 5 than does themi%ed stream. "herefore, the upfront lump/sum payment would e preferred. "hisconclusion differs from that in part c primarily due to the different patterns of cash flowassociated with the lump sum and mi%ed stream payment alternatives.

P3-0. Li%on huttleworth has een offered the choice among three retirement/planninginvestments. "he first investment offers a 5 percent return for the first 5 years, a 10 percentreturn for the ne%t 5 years, and a #0 percent return thereafter. "he second investment offers10 percent for the first 10 years and 15 percent thereafter. "he third investment offers aconstant 1# percent rate of return. Letermine, for each of the given numer of years, which

of these investments is the est for Li%on if he plans to ma>e one payment today into oneof these funds and plans to retire in the following numer of years2a. 15 years. #0 yearsc. &0 years



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Chapter 3 The Time Value of Money 7&

A3-0. a. nvestment M 12 uture alue actor 3 )1.05*5)1.10*5x )1.#0*53 5.115

nvestment M #2 uture alue actor 3 )1.10*10)1.15*53 5.#17

nvestment M &2 uture alue actor 3 )1.1#*153 5.+7+

f huttleworth retires in 15 years, investment M& has the highest future value

. nvestment M 12 uture alue actor 3 )1.05*5)1.10*5)1.#0*103 1#.7#7

nvestment M #2 uture alue actor 3 )1.10*10)1.15*103 10.+&

nvestment M &2 uture alue actor 3 )1.1#*#03 .6+6

f huttleworth retires in #0 years, investment M 1 has the highest future value

c. nvestment M 12 uture alue actor 3 )1.05*5)1.10*5 )1.#0*#03 79.90#

nvestment M #2 uture alue actor 3 )1.10*10)1.15*#03 +#.+51

nvestment M &2 uture alue actor 3 )1.1#*&03 #.5

f huttleworth retires in &0 years, investment M 1 has the highest future value.

Present Value of Cash Flow Streams

P3-1/. or the following :uestions, assume an end/of/year cash flow of $#50 and a 10 percentdiscount rate.a. =hat is the present value of a single cash flow!. =hat is the present value of a 5/year annuity!c. =hat is the present value of a 10/year annuity!d. =hat is the present value of a 100/year annuity!e. =hat is the present value of a $#50 perpetuity!

f. Lo you detect a relationship etween the numer of periods of an annuity and itsresemlance to a perpetuity!

A3-1/. a. 4 3 $#50 )1.10*/1 3 $##7.#7

. 4 3 $#50 1/ )1.10*/5 I 3 $+7.70 0.10

c. 4 3 $#50 1/ )1.10*/10 I 3 $1,5&6.1+

0.10

d. 4 3 $#50 1/)1.10*/100I3 $#,+.9#

0.10 e. 4 3 $#50 3 $#,500.00 0.10

f. As the numer of periods in an annuity increases )as we move from part to part d*and approaches infinity, the value of the annuity approaches the value of a perpetuity)part e*.

P3-11. Nog on to Hugh -houDs financial calculator we page)http2EEwww.interest.comEhughEcalcEsimple.org*, and loo> over the various calculator lin>savailale. ?efer ac> to some of the earlier time value prolems, and rewor> them with



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7+ Instructors Manual

these calculators. 'ow, run through several numerical scenarios to determine the impact ofchanging variales on your results.

A3-11. nternet e%ercise 8 answers will vary.

P3-12. Assume that you ;ust won the state lottery. Your pri@e can e ta>en either in the form of$+0,000 at the end of each of the ne%t #5 years )i.e., $1 million over #5 years* or as a lumpsum of $500,000 paid immediately.

a. f you e%pect to e ale to earn 5 percent annually on your investments over the ne%t#5 years, ignoring ta%es and other considerations, which alternative should you ta>e!=hy!

. =ould your decision in part )a* e altered if you could earn 7 percent rather than 5percent on your investments over the ne%t #5 years! =hy!

c. On a strict economic asis, at appro%imately what earnings rate would you eindifferent when choosing etween the two plans!

A3-12. 4An 3 4C"

1 8 1 I r )1 r* n

a. 4A#5 3 )$+0,000 E 0.05* 1 8 )1 .05*/#5 I

4A#5 3 $900,000 .70+67

4A#5 3 $56&,759At 5, ta>ing the award as an annuity is etter ecause its present value of $56&,579 islarger than the $500,000 lump/sum amount.

. 4A#5 3 )$+0,000 E 0.07*x 1 8 )1 .07*/#5I

4A#5 3 $571,+# .915751

4A#5 3 $+66,1++

At 7, ta>ing the award as a lump sum is etter ecause the present value of theannuity of $+66,1++ is less than the $500,000 lump/sum payment.

c.


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A3-15. 4 of deferred annuityQ 3 $5,000,000 1/)1.10* /&I )1.10*/&3 $,&+#,0++

.10Q 'ote the present value of the three deposits is measured at the eginning of year +, i.e.,

the end of year &.

P3-16. ?uth 'ail has ;ust received two offers for her seaside home. "he first offer is for $1 milliontoday. "he second offer is for an owner/financed sale with a payment schedule as follows2

End of Year Payment

0 (Today) $200,000

1 200,000

2 200,000

3 200,000

4 200,000

5 300,000

Assuming no differential ta% treatment etween the two options and that ?uth earns a rateof 9 percent on her investments, which offer should she ta>e!

A3-16. 4 of owner/financed sale2

n$ of

)ear *t+ Cash Flow G*1./+-t, Present Value0 $#00,000 G1.000000 3 $ #00,0001 #00,000 G.#5#6 3 195,196# #00,000 G.957&& 3 171,+69& #00,000 G.7&9 3 159,766+ #00,000 G.7&50&0 3 1+7,006

5 &00,000 G.69059& 3 #0+,175$1,066,601

?uth should ta>e the second offer for the series of payments ecause it has a higher presentvalue than the $1,000,000 payment today.

P3-17. Nandon Nowman, star :uarterac> of the university footall team, has een approachedaout forgoing his last two years of eligiility and ma>ing himself availale for theprofessional footall draft. "alent scouts estimate that Nandon could receive a signingonus of $1 million today along with a 5/year contract for $& million per year )payale atthe end of the year*. "hey further estimate that he could negotiate a contract for $5 millionper year for the remaining seven years of his career. "he scouts elieve, however, that

Nandon will e a much higher draft pic> if he improves y playing out his eligiility. f hestays at the university, he is e%pected to receive a $# million signing onus in two yearsalong with a 5/year contract for $5 million per year. After that, the scouts e%pect Nandon tootain a 5/year contract for $6 million per year to ta>e him into retirement. Assume thatNandon can earn a 10 percent return over this time. hould Nandon stay or go!



11/32


A3-17. 4 of Nandon entering the draft2igning onus 3 $ 1,000,000nitial contract 3 $&,000,000x 1/)1.10*/5 I 3 11,&7#,&60

.10

use:uent contract 3 $5,000,000x 1/)1.10*/7I )1.10*/5 3 15,11+,5#5 .10 4 3 $#7,+96,995

4 of Nandon playing out his eligiility2igning onus 3 $#,000,000x)1.10*/# 3 $ 1,65#,9&

nitial contract 3 $5,000,000x 1/)1.10*/5I )1.10*/# 3 15,66+,+09

.10

use:uent contract 3 $6,000,000x1/)1.10* /5I )1.10*/7 3 11,671,6&9

.10 4 3 $#9,99,&

ince the 4 of playing out his eligiility and then entering the draft is higher, Nandonshould stay in college.

P3-1. As part of your personal udgeting process, you have determined that in each of the ne%tfive years you will have udget shortfalls. n other words, you will need the amountsshown in the following tale at the end of the given year to alance your udgetKthat is,inflows e:ual outflows. You e%pect to e ale to earn 9 percent on your investments duringthe ne%t five years and wish to fund the udget shortfalls over these years with a singleinitial deposit.

n$ of )ear %u$&et Shortfall

1 $ 5,000# +,000& 6,000

+ 10,0005 &,000

a. How large must the lump/sum deposit e today into an account paying 9 percentannual interest to provide for full coverage of the anticipated udget shortfall!

. =hat effect would an increase in your earnings rate have on the amount calculated inpart a! %plain.

A3-1. a. n$ of %u$&et

)ear *t+ Shortfall G*1 ./+-t , Present Value

1 $5,000 G.#5#6 3 $ +,6&0 # $+,000 G .957&& 3 &,+# & $6,000 G.7&9 3 +,76& + $10,000 G .7&50&0 3 7,&50 5 $&,000 G.69059& 3 #,0+#

$ ##,#1+

An initial deposit of $##,#1+ would e needed to fund the shortfall for the patternshown in the tale.

. An increase in the earnings rate would reduce the amount calculated in part a."his edition is intended for use outside of the (.. only, with content that may e different from the (.. dition. "his may not e

resold, copied, or distriuted without the prior consent of the pulisher.


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P3-10. (se the following tale of cash flows to answer parts a / c. Assume an 9 percent discountrate.

End of Year Cash Flow

1 $10,000

2 10,000

3 10,000

4 12,000

5 12,000

6 12,000

7 12,000

8 15,000

9 15,000

10 15,000

a. olve for the present value of the cash flow stream y summing the present value ofeach individual cash flow.

. 'ow, solve for the present value y summing the present value of the three separateannuities )one current and two deferred*.

c. =hich method is etter for a long series of cash flows with emedded annuities!

A3-10. a. )ear Cash Flow Present Value 1 $10,000 $ ,#5 # 10,000 9,57& & 10,000 7,&9

+ 1#,000 9,9#0 5 1#,000 9,167

6 1#,000 7,56# 7 1#,000 7,00#

9 15,000 9,10+ 15,000 7,50+

10 15,000 6,+9"otal $7,977

. 4 3 $10,000 1/)1.09* /&I $1#,000 1/ )1.09* /+I )1.09*/& $15,000 1/)1.09* /&I

)1.09*/7

0.09 0.09 0.09 4 3 $#5,771 $&1,551 $##,556 4 3 $7,979

c. or a long series of cash flows with emedded annuities it is more efficient to calculateand sum the present values of the separate annuities as in part ."his >ind of prolemcan also e solved using the cash flow >eys of a financial calculator.

P3-2/. Jiven the mi%ed streams of cash flows shown in the following tale, answer parts )a* and)*2



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A3-21. a. 4 3 $#0,000 )1.01*/1# $#0,000 )1.01*/#+ $#0,000 )1.01*/&6 3 $+7,+7

. ffective rate 3 )1 .1#*1#8 1 3 1#.69 1#

4 3 $#0,000 )1.1#69*/1 $#0,000 )1.1#69*/# $#0,000 )1.1#69*/&3 $+7,+91

)rounding*

c. 4A&3 $#0,000 1 8 )1.1#69*/& I 3 $+7,+91 )rounding*

0.1#69

P3-22. Answer parts a8c for each of the following cases. Answer parts a8c for each of thefollowing cases.

Case

Amount

of Initial

Deposit

($)

Stated

Annual

Rate,r (%)

Compounding

Frequency,m

(times/year)

Deposit

Period

(years)A 2,500 6 2 5

B 50,000 12 6 3

C 1,000 5 1 10

D 20,000 16 4 6

a. -alculate the future value at the end of the specified deposit period.. Letermine the effective annual rate )!"*.c. -ompare the stated annual rate )r* to the effective annual rate)!"*. =hat relationship

e%ists etween compounding fre:uency and the stated and effective annual rates!

A3-22. a. -ompounding re:uency2 n3 4 )1 r*n

A. 5 3 $#,500 )1.0&*10


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A? 3 .05 3 5 A? 3 .17 3 17c. "he effective rates of interest rise with increasing compounding fre:uency.

P3-23. Pohn "ye has ;ust een hired as the new corporate finance analyst at /ll nterprises andhas received his first assignment. Pohn is to ta>e the $#5 million in cash received from a

recent divestiture and use part of these proceeds to retire an outstanding $10 million ondissue and the remainder to repurchase common stoc>. However, the ond issue cannot eretired for another two years. f Pohn can place the funds necessary to retire this $10million det into an account earning 6 percent compounded monthly#how much of the $#5million remains to repurchase stoc>!

A3-23. 4 of det oligation 3 $10,000,000 )1.005*/#+3 $9,971,957

unds remaining for stoc> repurchase 3 $#5,000,000 8 $9,971,957 3 $16,1#9,1+&.

P3-24. Pason pector has shopped around for the est interest rates for his investment of $10,000over the ne%t year. He has found the following2

Stated Rate Compounding

6.10% Annual

5.90% Semiannual

5.85% Monthly

a. =hich investment offers Pason the highest effective rate of return!. 'ow, assume that Pason wishes to invest his money for only si% months and the annual

compounded rate of 6.10 percent is not availale. =hich of the remaining investmentsshould Pason choose!

A3-24 a. 'om"nal ate Compoun$"n& ffet"e Annual ate

6.10 Annual 6.10

5.0 emiannual,,.51

#

5,.01

#

=+

5.95 Conthly01.61

1#

0595.1

1#

=+

"he annual/compounded rate of 6.10 is also the highest effective rate

. He would prefer the monthly compounding case ecause it offers a slightly highereffective annual rate of interest.

P3-25. "ara -utler is newly married and is now preparing a surprise gift of a trip to urope for herhusand on their tenth anniversary. "ara plans to invest $5,000 per year until thatanniversary and plans to ma>e her first $5,000 investment on their first anniversary. f sheearns an 9 percent rate on her investments, how much will she have saved for their trip ifthe interest is compounded in each of the following ways!a. Annually. Ruarterlyc. Conthly



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9# Instructors Manual

A3-25. a. ffective rate 3 nominal rate 3 9

A103 $5,000 )1.09*10 81 I 3 $7#,+&&

0.09

. ffective rate 3 )1 .09*+81 3 9.#+

+

A103 $5,000 )1.09#+*10 81 I 3 $7&,#6+

0.09#+

c. ffective rate 3 )1 .09*1# 8 1 3 9.&0 1#

A 3 $5,000 )1.09&*10 81 3 $7&,+7&

0.09&

P3-26. You plan to invest $#,000 in an individual retirement arrangement )?A* today at a statedinterest rateof 9 percent, which is e%pected to apply to all future years.a. How much will you have in the account at the end of 10 years if interest is

compounded as follows!)1* Annually)#* emiannually)&* Laily )assume a &60/day year*)+* -ontinuously

. =hat is the effective annual rate )!"* for each compounding period in part a!c. How much greater will your ?A account alance e at the end of 10 years if interest is

compounded continuously rather than annually!d. How does the compounding fre:uency affect the future value and effective annual rate

for a given deposit! %plain in terms of your findings in parts a8c.

A3-26. a. )1* 10 3 $#,000 )1.09*10 )#* 103 $#,000 )1.0+*

#0

10 3 $#,000 )#.15* 103 $#,000 )#.11*

10 3 $+,&19 10 3 $+,&9#

)&* 103 $#,000 )1.000##*&600 )+* 103 $#,000 e

.9

103 $#,000 )#.#09* 103 $#,000 )#.##6*

10 3 $+,+16 103 $+,+5#

. )1* A? 3 )1 .09E1*1

81 )#* A? 3 )1 .09E#*#

/1 A? 3 )1 .09*

1 / 1 A? 3 )1 .0+*#/ 1

A? 3 )1.09* 8 1 A? 3 )1.0916* / 1 A? 3 .09 3 9 A? 3 .0916 3 9.16

)&* A? 3 )1 .09E&60*&608 1 )+* A? 3 )e>8 1*

A? 3 )1 .000##*&608 1 A? 3 )e098 1*

A? 3 )1.09#+* 8 1 A? 3 )1.09&& 8 1* A? 3 .09#+ 3 9.#+ A? 3 .09&& 3 9.&&



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Chapter 3 The Time Value of Money 9&

c. "he ?A account alance at the end of 10 years will e $1&+ )$+,+5# 8 $+,&19* largerwith continuous rather than annual compounding.

d. "he more fre:uently interest is compounded at a given nominal annual rate, the greaterthe future value and the higher the effective annual rate, A?.

P3-27. "o supplement your planned retirement in e%actly +# years, you estimate that you need toaccumulate $##0,000 y the end of +# years from today. You plan to ma>e e:ual annualend/of/year deposits into an account paying 9 percent annual interest.a. How large must the annual deposits e to create the $##0,000 fund y the end of +#

years!. f you can afford to deposit only $600 per year into the account, how much will you

have accumulated y the end of the forty/second year!

A3-27. a. 4C" 3 A+# )1.09*+# 8 1 I . A+# 3 4C" )1.09*

+# 8 1 I

.09 .09

4C" 3 $##0,000 )&0+.#++* A+# 3 $600 )&0+.#++*

4C" 3 $7#&.10 A+# 3 $19#,5+6.+0

P3-2. You are planning to purchase a uilding for $+0,000, and you have $10,000 to apply as adown payment. You may orrow the remainder under the following terms2 a 10/year loanwith semiannual repayments and a stated interest rate of 6 percent. You intend to ma>e$6,000 payments, applying the e%cess over your re:uired payment to the reduction of theprincipal alance.a. Jiven these terms, how long )in years* will it ta>e you to fully repay your loan!. =hat will e your total interest cost!c. =hat would your interest cost e if you made no prepayments and repaid your loan y

strictly adhering to the terms of the loan!

A3-2. a. ?e:uired 4ayment2

4C" 3 $&0,000 3 $&0,000 3 $#,016.+7 1 8 )1 S.06E#T/# % 10I 1 8 )1.0&* /#0

.06 E# .0&

Amort"at"on She$ule

Per"o$

%e&"nn"n&

%alane Pament

8nterest

*./3 Pr"n"pal+ Pr"n"pal Prepa

n$"n&

%alane

1 $&0,000.00 $#,016.+7 $ 00.00 $1,116.+7 $&,9&.5& $#+,00.00# #+,00.00 #,016.+7 7+7.00 1,#6.+7 &,9&.5& 1,6+7.00

& 1,6+7.00 #,016.+7 59.+1 1,+#7.06 &,9&.5& 1+,#&6.+1+ 1+,#&6.+1 #,016.+7 +#7.0 1,59.&9 &,9&.5& 9,66&.505 9,66&.50 #,016.+7 #5.1 1,756.56 &,9&.5& #,#&.+16 #,#&.+1 #,016.+7 97.70 1,#9.77 +.6+ 0

$&,011.11"he loan will e paid off in 6 periods or & years.

. "otal interest cost 3 $&,011.11

c. "otal interest cost with no prepayments2"his edition is intended for use outside of the (.. only, with content that may e different from the (.. dition. "his may not e

resold, copied, or distriuted without the prior consent of the pulisher.


18/32

9+ Instructors Manual

#0 $#,016.+7 8 $&0,000 3 $10,.+0

P3-20. or e%cellent :ualitative discussions of the value of compounded interest on saving forfuture )retirement* oligations, see the following wesites2

http2EEwww.prudential.comEretirement )4rudential inancial*

http2EEwww.vanguard.com )"he anguard Jroup*

http2EEwww.fid/inv.com )idelity nvestments*

http2EEwww.loomerg.com ) out the financial calculator at


19/32


. -reate a schedule showing the cash inflows )including interest* and outflows of thisfund. How much remains on pencerDs twenty/first irthday!

A3-32. a. Amount needed at pencerDs 1&thirthday )in 10 years*2

3 $10,000 )1.09*/5 $11,000 )1.09*/6 $1#,000 )1.09*/7 $15,000 )1.09*/9

3 $6,905.9& $6,&1.97 $7,001.99 $9,10+.0& 3 $#9,9+&.61unding payment )1 .09*10 8 1 I 3 $#9,9+&.61

.09

unding payment 1+.+9656# 3 $#9,9+&.61

unding payment 3 $#9,9+&.61 3 $1,1.061+.+9656#

.

n$ of Spener9s

%"rth$a )ear

:epos"t

*;"th$rawal+

%e&"nn"n&

%alane

n$"n& %alane

*%e&"n %alane 1./+

+ $ 1,1.06 $ 1,1.06 $ #,150.&+5 1,1.06 +,1+1.+0 +,+7#.7#6 1,1.06 6,+6&.79 6,90.997 1,1.06 9,71.+ ,69.69 1,1.06 11,690.75 1#,615.#1 1,1.06 1+,606.#7 15,77+.79

10 1,1.06 17,765.9+ 1,197.1011 1,1.06 #1,179.16 ##,97#.+#1# 1,1.06 #+,96&.+9 #6,95#.561& 1,1.06 #9,9+&.6# &1,151.111+ 0 &1,151.11 &&,6+&.#015 0 &&,6+&.#0 &6,&&+.65

16 0 &6,&&+.65 &,#+1.+&17 0 &,#+1.+& +#,&90.7+19 )10,000* ,&90.7+ &+,71.#01 )11,000* #&,71.#0 #5,999.0#0 )1#,000* 1&,999.0 15,000.00#1 )15,000* 0

'othing remains in the account after the $15,000 withdrawal is made on pencerDs #1stirthday.

P3-33. Jo to the home page of the rate.com )http2EEwww.an>rate.com*, and otain currentaverage mortgage rates. =ith this information, go to Hugh -houDs mortgage calculator

)http2EEwww.interest.comEhughEcalcEsimple.org*. 4rovide the re:uested variales to createan amorti@ation schedule. 'ow, re/create the schedule with different prepayment amounts.=hat impact does the prepayment have on total interest and the term of the loan!

A3-33. nternet e%ercise 8 answers will vary.

P3-34. Cary ullivan, capital outlay manager for =a%y =idgets, has een instructed to estalish acontingency fund to cover the e%penses over the ne%t two years )#+ months* associatedwith repairing defective widgets from a new production process. =a%y =idgetsD controllerwants to ma>e e:ual monthly cash deposits into this fund. f Cary faces the following



20/32


monthly repair costs and has $1 million to start the fund today, what will e her monthlypaymentsinto the fund in order to assume that all repair costs will e covered! Cary willma>e her first payment one month from today, and the fund will earn 6 percent,compounded monthly.

Months Repair Costs per Month14 $500,000

512 $250,000

1324 $100,000

A3-34. 4resent alue of uture Niailities 3 4resent alue of ?e:uired unding Annuity

)rate 3 .06E1# 3 .005F M periods 3 1# n*

$500,000 1 8 )1.005*/+I

.005

$#50,000

1 8 )1.005*

/9

I

)1.005*

/+

3 Annuity

18 )1.005*

/#+

I $1,000,000 .005 .005

$100,000 1 8 )1.005*/1# I )1.005*/1#

.005

$+,96,751 3 Annuity ##.56# 1,000,000

$&,96,750 3 Annuity ##.56#

Annuity 3 $176,65

P3-35. (se a spreadsheet to create amorti@ation schedules for the following five scenarios. =hat

happens to the total interest paid under each scenario!a. cenario 12

Noan amount2 $1 millionAnnual rate2 5 percent"erm2 &60 months4repayment2 $0

. cenario #2 ame as 1, e%cept annual rate is 7 percentc. cenario &2 ame as 1, e%cept term is 190 monthsd. cenario +2 ame as 1, e%cept prepayment is $#50 per monthe. cenario 52 ame as 1, e%cept loan amount is $1#5,000

A3-35. a.

%e&"nn"n& n$"n&Per"o$ %alane Pament 8nterest Pr"n"pal %alane

1 $1,000,000.00 $5,&69.## $+,166.67 $1,#01.55 $9,79.+5# $9,79.+5 $5,&69.## $+,161.66 $1,#06.56 $7,51.9& $7,51.9 $5,&69.## $+,156.6& $1,#11.59 $6,&90.&1

QQQQ Years in etween are not shown for practical purposes QQQQ

&59 $15,71.&7 $5,&69.## $66.55 $5,&01.67 $10,66.70



21/32


&5 $10,66.70 $5,&69.## $++.+6 $5,&.76 $5,&+5.+&60 $5,&+5.+ $5,&69.## $##.#7 $5,&+5.+ $0.00

"otal $,557.9+ $1,000,000.00.

%e&"nn"n& n$"n&

Per"o$ %alane Pament 8nterest Pr"n"pal %alane1 $1,000,000.00 $6,65&.0# $5,9&&.&& $91.6 $,190.&1# $,190.&1 $6,65&.0# $5,9#9.55 $9#+.+7 $9,&55.9+& $9,&55.9+ $6,65&.0# $5,9#&.7+ $9#.#9 $7,5#6.55


&59 $1,7#9.+6 $6,65&.0# $115.09 $6,5&7.+ $1&,10.5#&5 $1&,10.5# $6,65&.0# $76.+ $6,576.09 $6,61+.++&60 $6,61+.++ $6,65&.0# $&9.59 $6,61+.++ $0.00

"otal $1,&5,099.9 $1,000,000.00

c.%e&"nn"n& n$"n&

Per"o$ %alane Pament 8nterest Pr"n"pal %alane

1 $1,000,000.00 $7,07.+ $+,166.67 $&,7+1.#7 $6,#59.7 $6,#59.7& $7,07.+ $+,151.09 $&,756.96 $#,501.97

& $#,501.97 $7,07.+ $+,1&5.+# $&,77#.51 $99,7#.&6


179 $#&,5#7.+7 $7,07.+ $9.0& $7,90.1 $15,717.5717 $15,717.57 $7,07.+ $65.+ $7,9+#.+5 $7,975.1#190 $7,975.1# $7,07.+ $.91 $7,975.1# $0.00

"otal $+#&,+#9.5& $1,000,000.00

d.

%e&"nn"n& n$"n&

Per"o$ %alane Pament 8nterest Pr"n"pal Prepa %alane

1 $1,000,000.00 $5,&69.## $+,166.67 $1,#01.55 $#50.00 $9,5+9.+5# $9,5+9.+5 $5,&69.## $+,160.6# $1,#07.60 $#50.00 $7,00.95& $7,00.95 $5,&69.## $+,15+.55 $1,#1&.67 $#50.00 $5,6#7.19

QQQQ Years in etween are not shown for practical purposes QQQQQQQ =ith the prepayment of $#50.00 per month youDll have paid the loan in full y month QQQ

+ $1&,977.& $5,&69.## $57.9# $5,&10.& $#50.00 $9,&17.00 $9,&17.00 $5,&69.## $&+.65 $5,&&&.56 $#50.00 $#,7&&.+& $#,7&&.+& $5,&69.## $11.& $#,7&&.+& $0.00 $0.00

"otal $9#9,665.10 $19,750.00 $91,#50.004rincipal 4repay $1,000.000.00

e.%e&"nn"n& n$"n&

Per"o$ %alane Pament 8nterest Pr"n"pal %alane



22/32


1 $1#5,000.00 $671.0& $5#0.9& $150.1 $1#+,9+.91# $1#+,9+.91 $671.0& $5#0.#1 $150.9# $1#+,69.& $1#+,69. $671.0& $51.59 $151.+5 $1#+,5+7.5+


&59 $1,6.+# $671.0& $9. $66#.71 $1,&&&.71&5 $1,&&&.71 $671.0& $5.56 $665.+7 $669.#+&60 $669.#+ $671.0& $#.79 $669.#+ $0.00

"otal $116,56.7& $1#5,000.00

P3-36. Letermine the annual payment re:uired to fund a future annual annuity of $1#,000 peryear. You will fund this future liaility over the ne%t five years, with the first payment tooccur one year from today. "he future $1#,000 liaility will last for four years, with thefirst payment to occur seven years from today. f you can earn 9 percent on this account,how much will you have to deposit each year over the ne%t five years to fund the futureliaility!

A3-36. 4resent alue of uture Niaility 3 4resent alue of unding Annuity

1 8 )1.09*/+I )1.09*/6 $1#,000 3 1 8 )1.09*$%I unding annuity

.09 .09

$#5,0+6.+# 3 &.#71 unding annuity

unding annuity 3 $6,#7&.0+

A$$"t"onal Appl"at"ons of "me-Value ehn"y and is not concerned aout their differing lives. Her decision will e asedsolely on the rate of return she will earn on each annuity. "he >ey terms of each of the fourannuities are shown in the following tale.

Annu"t

Prem"um

Pa"$ o$a

Annual

%enef"t

L"fe

*ears+

A $&0,000 $&,100 #0< #5,000 &,00 10- +0,000 +,#00 15L &5,000 +,000 1#

a. -alculate to the nearest 1 percent the rate of return on each of the four annuities Pill isconsidering.

. Jiven PillDs stated decision criterion, which annuity would you recommend!

A3-37. a. Noan A Noan e a deposit at the end of this year in an account that pays 6 percentcompounded annually, and an identical deposit at the end of each year, with the last depositoccurring when he turns 19. f an annual deposit of $1,+9+ will allow you to reach yourgoal, how old is your son now!

A3-4/. 4 of funding annuity 3 4 of college e%pense annuity

$1,+9+ 1/ )1.06* /nI 3 $+,000 1 8 )1.06* /+I )1.06*/n 81I

.06 .06

$1,+9+ 1/)1.06* /nI 3 $1&,960.+# )1.06*/n/1I

.06

ia trial and error or a financial calculator, solve for n 3 9."hus, your son will turn 19 eight years from today and is 10 years old now.

P3-41. Letermine the length of time re:uired to doule the value of an investment, given thefollowing rates of return.a. + percent. 10 percentc. &0 percentd. 100 percent

A3-41. a. 3 4 )1 r*n

#.0 3 )1.0+*n

log# 3 nlog1.0+

0.&01 3 n 0.017

n 3 17.7 years

. #.0 3 )1.1* n

log# 3 nlog1.1

0.&01 3 n 0.0+1+

n 3 7.#7 years

c. #.0 3 )1.&'n

log# 3 nlog1.&

0.&01 3 n 0.11&

n 3 #.6+ years

d. #.0 3 )#'n

n 3 1 year



26/32

# Instructors Manual

P3-42. -onsider the following three investments of e:ual ris>. =hich offers the greatest rate of

return!

8nestment

n$ of )ear A % C

0 /$10,000 /$#0,000 /$#5,0001 0 ,500 #0,000# 0 ,500 &0,000& #+,600 ,500 /1#,600

A3-42. 'ote that all returns were calculated using a financial calculator.

?eturn on nvestment A2

$#+,600 3 $10,000 ) 1 r*&

r3 &5

?eturn on nvestment


27/32

Chapter 3 The Time Value of Money &

A3-43. a. 4 3 n( )

+ nr11

.-ase -ase

A 4 3 +4r,yrs A ame value $500 3 $900 4r, + yrs

.6#5 3 4r,+ yrs1# X r X 1&-alculator olution2 1#.+7

< 4 3 64r, 6 yrs < ame value

$1,500 3 $1,50 4r, 6 yrs..76 3 4r, 6 yrs.+XrX5-alculator solution2 +.+7

- 4 3 4r, - ame value$#,#90 3 $#,500 4r, yrs..1# 3 4r, yrs.1XrX#-alculator solution2 1.0&

c. "he growth and the interest rate should e e:ual, ecause they represent the samething.

P3-44. Letermine which of the following three investments offers you the highest rate of return onyour $1,000 investment over the ne%t five years.

nvestment 12 $#,000 lump sum to e received in five yearsnvestment #2 $&00 at the end of each of the ne%t five yearsnvestment &2 $#50 at the eginning of each of the ne%t five years

a. =hich investment offers the highest return!. =hich offers the highest return if the payouts are douled )i.e., $+,000, $600, and

$500*!c. =hat causes the ig change in the returns on the annuities!

A3-44. a. ?eturn on nvestment M 12

$#,000 3 $1,000 )1 r*5

#.0 3 )1 r *5)#.0*1E5 3 1 r1.1+97 3 1 rr = 1+.97

?eturn on nvestment M #2

$1,000 3 $&00 1 1 8 1 I

r )1r* n



28/32

+ Instructors Manual

ia trial and error or a financial calculator2 r 15.#+

?eturn on nvestment M &2

$1,000 3 $#50 1 1 8 1 I )1 r*

r )1 r* n

ia trial and error or a financial calculator2 r1#.5

"hus, nvestment M # has the highest return.

. ?eturn on nvestment M 12

$+,000 3 $1,000 )1 r*5

r 3 &1.5

?eturn on nvestment M#2

$1,000 3 600 1 1 8 1 I

r )1 r* n

r5#.90

?eturn on nvestment M &2

$1,000 3 $500 1 1 8 1 I )1 r*

r )1 r* n

r

#.76

"hus, nvestment M & has the highest return.

c. "he annuities have much greater sensitivity ecause their intermediate cash flows arereinvested whereas the lump sum investment does not generate any intermediate cashflows that can e reinvested.

P3-45. ind the rates of return re:uired to do the following2a. Loule an investment in + years. Loule an investment in 10 yearsc. "riple an investment in + years

d. "riple an investment in 10 years

A3-45. 3 4 )1 r*+

a. #.0 3 )1 r*+

)#* 3 )1 r*1.19#07 3 1 r

r19.#



29/32


. #.0 3 )1 r* 10

)#*1E103 )1 r*1.07177& 3 1 r

r 7.19

c. &.0 3 )1 r'+

)&*1E+ 3 )1 r*1.&1607 3 1 r

r &1.61

d. &.0 3 )1 r'10

)&*1E10 3 )1 r*1.1161#& 3 1 r

r3 11.61

P3-46. "he viatical industry offers a rather grim e%ample of present value concepts. A firm in thisusiness, called a viator, purchases the rights to the enefits from a life insurance contractfrom a terminally ill client. "he viator may then sell claims on the insurance payout toother investors. "he industry egan in the early 10s as a way to help AL patientscapture some of the proceeds from their life insurance policies for living e%penses.

uppose a patient has a life e%pectancy of 19 months and a life insurance policy with adeath enefit of $100,000. A viator pays $90,000 for the right to the enefit, and then sellsthat claim to another investor for $90,500.a. rom the point of view of the patient, this contract is li>e ta>ing out a loan. =hat is the

compound annual interest rate on the loan if the patient lives e%actly 19 months! =hatif the patient lives &6 months!

. rom the point of view of the investor, this transaction is li>e lending money. =hat is

the compound annual interest rate earned on the loan if the patient lives 19 months!=hat if the patient lives ;ust 1# months!

A3-46. a. 90,000 3 100,000E)1r*1.5

r 3 16, if the patient lives 19 months

90,000 3 100,000E)1r*&

r 3 7.7, if the patient lives &6 months

. 90,500 3 100,000E)1r*1.5

r 3 15.75, if the patient lives 19 months

90,500 3 100,000E)1r*r 3 #+.&, if the patient lives 1# months



30/32


=>!S>' >' %us"ness Shool $"t"on? ince P3-47and P3-4are ased on using a livedata ase, answers will vary from moment to moment. "his is a chance for your students to use aversion of a tool that -As use every day. f you would li>e to use "homson O' ut are unsureaout how to proceed, contact your rep for copies of! (uide to )sing Thomson *ne +,y

?osemary -arlson at Corehead tate (niversity. Her guide will lead you through this powerfultool availale to your students with the purchase of any new copy of this te%t.

Answer to MiniCase

Present Value

-asino.com -orporation is uilding a $#5 million office uilding in Nas egas and is financing theconstruction at an 90 percent loan/to/value ratio, where the loan is in the amount of $#0,000,000."his loan has a ten/year maturity, calls for monthly payments, and is contracted at an interest rateof 9 percent.

Ass"&nment(sing the aove information, answer the following :uestions.1. =hat is the monthly payment!#. How much of the first payment is interest!&. How much of the first payment is principal!+. How much will -asino.com -orporation owe on this loan after ma>ing monthly payments for

three years )the amount owed immediately after the thirty/si%th payment*!5. hould this loan e refinanced after three years with a new seven/year 7 percent loan, if the

cost to refinance is $#50,000! "o ma>e this decision, calculate the new loan payments and thenthe present value of the difference in the loan payments.

6. ?eturning to the original ten/year 9 percent loan, how much is the loan payment if thesepayments are scheduled for :uarterly rather than monthly payments!

7. or this loan with :uarterly payments, how much will -asino.com -orporation owe on thisloan after ma>ing :uarterly payments for three years )the amount owed immediately after thetwelfth payment*!

9. =hat is the annual percentage rate on the original ten/year 9 percent loan!. =hat is the effecti-e annual rate !"'on the original ten/year 9 percent loan!



31/32


Answers

1. 4 3 #0,000,000

n 3 10 1# 3 1#0

i 3 9E1# 3 .6667

Conthly payment 3 4C" 3 $#+#,655.1

#. nterest 3 Noan amount Conthly interest rate

3 $#0,000,000 9E1#

3 $1&&,&&&.&&

&. 4rincipal first month 3 Conthly payment 8 nterest first month3 $#+#,655.1 8 $1&&,&&&.&&3 $10,.96

+. 7 Years 1# Conths 3 9+ 4ayments remaining

4C" 3 $#+#,655.1n 3 9+i 3 9E1# 3 .66674rincipal after & years 3 4 3 $15,569,577.6#

5. 'ew loan payments24 3 $15,569,577.6#

n 3 7 Years 1# Conths 3 9+ Conths

i 3 7E1# 3 .59&&4C" 3 $#&+,71.55

Lifference in loan payments 3 $#+#,655.1 8 $#&+,71.55 3 $769&.6&

4resent value of difference24C" 3 $7,69&.6&

n 3 7 Years 1# Conths 3 9+

i 3 7E1# 3 .59&&4 3 $50,06.+9

ince savings from refinancing is greater than the cost to refinance )$50,06.+9 Z $#50,000*,then the loan should e refinanced.

6. 4 3 $#0,000,000

n 3 10 Years + Ruarters 3 +0 Ruarters

i 3 9E+ 3 #Ruarterly payments 3 4C" 3 $7&1,11+.6

7. n 3 7 Years + Ruarters 3 #9

i 3 9E+ 3 #4C" 3 $7&1,11+.6Amount owed after & years 3 4 3 $15,55,056.50

9. Annual percentage rate 3 9



32/32


. A? 3 )1 r *m 8 1

#70&.+*055.1)

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